Monte Carlo (MC) sampling algorithms are an extremely widely-used technique to estimate expectations of functions f(x), especially in high dimensions. Control variates are a very powerful technique to reduce the error of such estimates, but in their conventional form rely on having an accurate approximation of f, a priori. Stacked Monte Carlo (StackMC) is a recently introduced technique designed to overcome this limitation by fitting a control variate to the data samples themselves. Done naively, forming a control variate to the data would result in overfitting, typically worsening the MC algorithm's performance. StackMC uses in-sample / out-sample techniques to remove this overfitting. Crucially, it is a post-processing technique, requiring no additional samples, and can be applied to data generated by any MC estimator. Our preliminary experiments demonstrated that StackMC improved the estimates of expectations when it was used to post-process samples produces by a "simple sampling" MC estimator. Here we substantially extend this earlier work. We provide an in-depth analysis of the StackMC algorithm, which we use to construct an improved version of the original algorithm, with lower estimation error. We then perform experiments of StackMC on several additional kinds of MC estimators, demonstrating improved performance when the samples are generated via importance sampling, Latin-hypercube sampling and quasi-Monte Carlo sampling. We also show how to extend StackMC to combine multiple fitting functions, and how to apply it to discrete input spaces x.

Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using randomly generated samples of the function. In light of the increasing interest in uncertainty quantification and robust design applications in aerospace engineering, the calculation of expected values of such functions (e.g. performance measures) becomes important. However, MC techniques often suffer from high variance and slow convergence as the number of samples increases. In this paper we present Stacked Monte Carlo (StackMC), a new method for post-processing an existing set of MC samples to improve the associated integral estimate. StackMC is based on the supervised learning techniques of fitting functions and cross validation. It should reduce the variance of any type of Monte Carlo integral estimate (simple sampling, importance sampling, quasi-Monte Carlo, MCMC, etc.) without adding bias. We report on an extensive set of experiments confirming that the StackMC estimate of an integral is more accurate than both the associated unprocessed Monte Carlo estimate and an estimate based on a functional fit to the MC samples. These experiments run over a wide variety of integration spaces, numbers of sample points, dimensions, and fitting functions. In particular, we apply StackMC in estimating the expected value of the fuel burn metric of future commercial aircraft and in estimating sonic boom loudness measures. We compare the efficiency of StackMC with that of more standard methods and show that for negligible additional computational cost significant increases in accuracy are gained.